Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #213 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×D4⋊(C3⋊C8) |
| Sylow | C3 ⋊ C2×D4⋊C8 |
344 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 21 points:
(3,4,5)(6,7)(8,13)(9,11)(10,12);(4,5)(7,10)(8,12)(11,13)(14,15,16,18,17,19,20,21);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D4⋊(C3⋊C8), Dic3×D4⋊C8, #10685789, … |
| maximal normal subgroup of | C4×D4⋊(C3⋊C8), Dic3×D4⋊C8, #10685789, … |
| maximal quotient of | C2×#67639, C2×#67638, #10683126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4172);